# Active Ornstein-Uhlenbeck particles

## TL;DR

This paper investigates the equilibrium properties of Active Ornstein-Uhlenbeck particles, showing that they can be in equilibrium only under specific conditions and analyzing how small persistence times affect their probability densities.

## Contribution

It proves that AOUPs are in equilibrium only if their interaction potential has zero third derivatives and derives the effects of small persistence time on their stationary distributions.

## Key findings

- AOUPs are in equilibrium only if the potential's third derivatives are zero.
- Small persistence time leads to a local equilibrium in momenta and a diffusion equation for position.
- Higher-order terms in persistence time break detailed balance, indicating non-equilibrium behavior.

## Abstract

Active Ornstein-Uhlenbeck particles (AOUPs) are overdamped particles in an interaction potential subject to external Ornstein-Uhlenbeck noises. They can be transformed into a system of underdamped particles under additional velocity dependent forces and subject to white noise forces. There has been some discussion in the literature on whether AOUPs can be in equilibrium for particular interaction potentials and how far from equilibrium they are in the limit of small persistence time. By using a theorem on the time reversed form of the AOUP Langevin-Ito equations, I prove that they have an equilibrium probability density invariant under time reversal if and only if their smooth interaction potential has zero third derivatives. In the limit of small persistence Ornstein-Uhlenbeck time $\tau$, a Chapman-Enskog expansion of the Fokker-Planck equation shows that the probability density has a local equilibrium solution in the particle momenta modulated by a reduced probability density that varies slowly with the position. The reduced probability density satisfies a continuity equation in which the probability current has an asymptotic expansion in powers of $\tau$. Keeping up to $O(\tau)$ terms, this equation is a diffusion equation, which has an equilibrium stationary solution with zero current. However, $O(\tau^2)$ terms contain fifth and sixth order spatial derivatives and the continuity equation no longer has a zero current stationary solution. The expansion of the overall stationary solution now contains odd terms in the momenta, which clearly shows that it is not an equilibrium.